Sprouts is a game contrived by J. H. Conway and M. S. Paterson in the 1960s.
It is an impartial game with for two players played on the a plane with some spots. Each move consists of both
- joining two spots (could be the same spot) with a simple curve which does not go through existing spots or curves, such that the degree of each spot after the move does not exceed 3; and
- placing a new spot on that curve.
The winner/loser is who
Who makes the last move is the winner/loser according to normal/misère play convention.
This game is of topological nature but there are only finitely many inequivalent options at each move, and the game always terminates after finitely many moves (in fact, bounded by number of initial spots), making it an combinatorial game.
It enjoys some popularity, as reflected by the existence of a world association (the WGOSA, World Game of Sprouts Association).
Here is a website dedicated to the determination of the theoretical winnerwinners. A pattern with period 6 emerged under both play conventions. The researchers have published several papers and even considered Sprouts on general surfaces ("compact" is not essential, I think), and proved that the theoretical winner of the Sprouts game with a fixed number of spots on different compact surfaces is ultimately periodic in genus, with period 1/period 2 in the case of orientable/non-orientable surfaces.